Embed media option for Fbx export
There is an option to do that, actually. 
Make sure you're in Blender Render (internal) render mode, though, otherwise it won't work, in my experience.
In Fbx export options, under Main, look all the way down at "Path mode". Set that to "Copy" and make sure to check the little file-in-a-box button to the right (called "Embed textures").